Dynamic Analysis Of Delayed Markovian Switching Complex-valued Neural Networks

Complex-valued neural networks(CVNNs)are the kind of networks that deal with infor-mation in the complex plane,where their state variables,connection weight matrices,activation functions and external inputs are all complex-valued.The weight multiplication at synapses by CVNNs will yield the phase rotation as well as the amplitude amplification or attenuation,hence,they have superior advantages in ability of network learning and self-organization.On the other hand,CVNNs have the advantage to solve certain problems that cannot be solved by their real-valued counterparts,such as the XOR problem and the symmetry detection prob-lem.After several decades of development,CVNNs have been widely used in complex signal processing,optoelectronics,gray-scale image processing,neural cryptography,radar imaging,associative memory,etc.From the perspective of dynamics,CVNNs are a class of complex nonlinear dynamic systems,and many applications of neural networks rely on the intensive study of their dynamic behaviors.In addition,in the actual network transmission process,due to the limited switching speed of the amplifier,the communication time between the neurons is also limited,which inevitably causes delay phenomenon in the neuron information transmis-sion.The appearance of the time delay will greatly affect the performance of the system,and even lead to oscillation,bifurcation,instability,etc.In the actual engineering fields,due to the interference of environmental noises and the limitations of physical factors,the system structure and parameters are always severely af-fected,which will greatly influence performances of the system.Markovian switching neural network,as a special hybrid system,can vividly describe this kind of systems whose structure and parameters are subject to changes.In the past few decades,the main research results in this field focus on the stability analysis,filtering and fault detection of stochastic system-s,and control synthesis.However,the research on delayed Markovian switching CVNNs is still on the initial stage,and the existing relevant results are not enough,which implies that there are lots of challenging problems to be solved urgently.The purpose of this disserta-tion is to discuss the dynamics of several types of delayed Markovian switching CVNNs with network-induced complexities,which include the Markovian switching CVNNs,the piecewise-homogeneous Markovian switching CVNNs,the semi-Markovian switching CVNNs,and the hidden Markovian switching models.The network-induced complexities taken into account mainly include impulse,randomly occurring uncertainty,missing measurements,and random-ly occurring nonlinearity.Furthermore,some novel inequalities are proposed in the complex number field,for example,the complex-valued fundamental inequality and the complex-valued reciprocal convex inequality.Based on these,attention of this dissertation primarily focus-es on the dynamical behaviors such as(exponential)stability,synchronization,stabilization,dissipativity,passivity,H_?estimation and non-fragile asynchronous state estimation.More specifically,the main content of this dissertation is organized as follows.The second chapter investigates the global exponential stability and synchronization prob-lems for Markovian switching CVNNs under the impulsive control.Under two types of activa-tion functions,by utilizing the matrix measure approach,the Lyapunov stability theory and the impulsive differential inequality,several sufficient criterion are firstly derived to ensure the global exponential stability of the considered impulsive CVNN,and the exponential conver-gence rate is also presented.Subsequently,the exponential synchronization problem of the impulsive CVNNs is analyzed through proposing an appropriate mode-dependent controller,and the obtained theoretical results are easy to be verified and implemented in practice.The third chapter is concerned with the stability and stabilization problems for delayed Markovian switching CVNNs with incomplete transition rates.Firstly,a class of Markovian switching CVNNs with mixed delays and partly unknown transition rates are considered.Several sufficient mode-dependent conditions are established to ensure the global asymptotic mean-square stability of the addressed networks by utilizing the Lyapunov stability theory,the stochastic analysis technique and the properties of the transition rate matrix.Furthermore,when the system is unstable,in order to achieve the stabilization effect,a mode-dependent feed-back controller is designed,which is memoryless.Secondly,a class of piecewise-homogeneous Markovian switching CVNNs with mode-dependent delays and incomplete transition rates are considered.Here,the time-varying delay is piecewise-homogeneous Markovian mode-dependent.By constructing an appropriate Lyapunov functional,some sufficient criteria are obtained to guarantee the global exponential mean-square stability of the considered network.In addition,when it comes to the stabilization problem,a memory mode-dependent feedback controller is effectively designed.The fourth chapter analyzes the dissipativity and passivity problems for Markovian switch-ing CVNNs with randomly occurring uncertainties and general uncertain transition rates.Among them,the randomly occurring uncertainties are characterized by certain mutually independent Bernoulli-distributed white sequences,and the transition rates are generally un-certain,that is,completely unknown or unknown but with known upper/lower bounds.By closely combining with the generalized It(?) formula in the complex domain,the robust analysis technique and the stochastic analysis method,some mode-dependent criteria are achieved to ensure the stochastic Markovian switching CVNNs to be robustly dissipative/passive in the sense of expectation.In addition,comparisons with different criteria in the literature on dis-sipativity/passivity analysis are presented.It can be concluded that stochastic factors,i.e.,the Markovian switching mechanism and the Brownian motion,have significant effect on the dissipativity/passivity performance index.The fifth chapter considers the H_?estimation problem for stochastic semi-Markovian switching CVNNs subject to incomplete measurement outputs,where the time-varying delay also depends on another semi-Markovian process.A sequence of random variables taking values in the interval[0,1]is introduced to depict the phenomenon of missing measurements.Based on the generalized It(?)'s formula in complex form with semi-Markovian switching,the complex-valued reciprocal convex inequality as well as some stochastic analysis methods,some mode-dependent sufficient conditions are derived,under which the estimation error system is exponentially mean-square stable with a given H_?disturbance attenuation level.On this basis,rationality of the estimator design scheme and effectiveness of the obtained results are further demonstrated.The sixth chapter discusses the non-fragile asynchronous state estimation problem for discrete Markovian switching complex-valued networks subject to randomly occurring nonlin-earities and partly accessible mode detection.In the same way,a sequence of random variables with values in the interval[0,1]is used to describe the phenomenon of randomly occurring nonlinearities.By resorting to the hidden Markovian switching model,a novel non-fragile asynchronous mode-dependent state estimator is designed.Based on the Lyapunov functional theory,the complex-valued reciprocal convex inequality and the stochastic analysis technique,the issue of global asymptotic mean-square stability for the augmented estimation error system is dealt with.Whereafter,the asynchronous estimator matrices can be determined in terms of feasible solutions of the obtained complex matrix inequalities.It is noteworthy that,in the practical applications,due to the possibly unobservable system modes or high costs in mea-suring the mode information,this chapter assumes that the mode detection probabilities are partly accessible,which infers that the research here is more general and thus more convenient for practical applications.